Let us examine this one section (9:35-9:38) of only a few seconds of a video of Okamoto-sensei, a Daito-Ryu practitioner, demonstrating a way to free oneself from an opponent grabbing both your wrists when both are in a seated position. He shows one of the most fundamental techniques of daito-ryu, in which he raises his hands in a circular motion towards the shoulders of the opponent (or towards the sky) and then sends the opponent to the side. The opponent appears to be pushed back through the arms before being jerked forward and rolling out. Additionally, the seated practitioner's movements throughout are small and subtle.

6 Answers
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The other answers are good ones, and all bring up valid points. I think they're all a bit long though. I like stslavik's answer (it's very technical and detailed, which is nice), but I think he missed a couple points:

The first is that Okamoto-sensei isn't letting his uke settle into the grab. As soon (or before) their hands have touched his wrists, he's moving into the technique. This difference in timing can make or break a technique.

The second is that while the students may not be deliberately 'going with the technique', they are almost certainly not attacking. They are giving Okamoto-sensei an attack; not actually attacking him. Another important difference, in that the students are more willing to give up their position when they haven't fought to get it.

Lastly, if there's one part that is really the key to this technique, it's that Okamoto-sensei puts himself underneath (not literally, but focus/force/positional-wise) the uke's elbows and then applies force; this creates a joint lock and from there, any sort of pin or throw can be applied.

There is no one, unifying piece that is at work in the video provided, but many pieces working in parallel. I will be as explicit in naming as many as possible to give an accurate view of the principles at work.

Balance – Okamoto-sensei (I have it as Okamoto Seigo, though have seen Okamoto Shogo as well, especially from Russian videos) sits upright in seiza, shoulders over hips. Uke sites in seiza at a distance of about a foot; in order for uke to reach sensei's wrists, he has to give up his balance – that is, he breaks alignment of his hips and shoulders, whereas sensei need not do so. Therefore, uke is imbalanced, his arms at a full extension by virtue of needing to keep some balance (avoiding falling forward). In this way, uke gives himself to sensei, and sensei needs only to steer him right or left to point him where to fall. This loss of balance, interestingly enough, occurs at 9:35-9:36, immediately following uke's best attempts to seat himself in a balanced posture.

Locking – Once that uke makes contact, he is supported partly by sensei. Sensei now has a lever in the form of uke's arms provided that he can maintain rigidity. To do this, sensei first raises the hands and points up (an exaggerated example of this technique is available on the DVDs Daito-ryu Roppokai and Daito-ryu Roppokai Hawaii Seminar, the latter being from which the question's link is drawn.) which bends the wrist, locks the elbows, and pushes the arms up into the shoulders. So long as sensei maintains his balance, the pressure on the joints will prevent the lock from being released. This occurs at 9:37 in the video.

Flow – Called nagare in Japanese, or known as The Wave in Systema Ryabko, the continuity of motion from 9:36 to 9:38 begins in the knees, a slight wave moving up from the knees through the hips and up through the arms. This combined with the lock setting at 9:37 lifts uke, and the continued movement in a circle biased toward one side leads uke's already natural tendency forward.

Will – This is, without a doubt, the most difficult aspect to teach. Without getting too philosophical, the martial artist has to realize that we are not all that different than Hermetics; we are workers of will. When we move, we move where we are invited and take the space that we desire. We command our movements take priority. The novice struggles with technique, assuring himself he's performing correctly when it works, never understanding why it just so happens to always work for sensei. The reason is that the senseiwills for the technique to work, consciously or not. His presence, his actions, and his authority all lend themselves to his ability to act.

On the simplest of levels, I understand that if I tell you, "Don't think of a black cat." you must, in order to understand me, think of a black cat. By making this statement, I am subjecting you to my will, whether it is your intent to be subjected or not. It is unavoidable. For a technique like the te hodoki seen in the video to work, the martial artist must first bend the reality around him; that is, he must create the illusion that the only viable option is for the uke to grab his wrists. To say that the martial artist must wait for that opportunity to present itself is misguided, as that would put him into the realm of reaction, which is too slow to be effective. Why does uke grab for sensei's wrists then? Because sensei, through his position and his intention has set up the scenario in which that is the appropriate response. Uke, then, puts himself into position such that he has no choice but to be thrown. [ Related - Reality and Non-Reality of Techniques]

I am, of course, asking you to accept this conclusion based on my experiences knowing full well that you do not have the same experiences necessary to have come to these conclusions as well. You'll notice in here that I only once mentioned a technique (calling what was performed as "te hodoki), but gave you four points to examine. In order for you to make use of what I'm telling you, you're going to have to take a moment to consider that techniques are not the smallest unit of training. These are all principles, the atomic units upon which techniques, forms, and even movement are all based. If there's anything at all that you feel remotely unclear upon, please ask, and I will do my best to break it down further.

This description fits my understanding of a parlor trick, as described in my answer.
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Dave LiepmannMay 14 '12 at 21:54

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Coming from a background as a magician, and having a long background of martial arts training, I'm quite confident that what I describe isn't a parlor trick; these same principles underly every technique in martial arts in one way or another, even the ones you train in.
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stslavikMay 15 '12 at 15:47

One is the technique that Okamoto-sensei is performing, the other is the technique that the uke is performing. It is important to observe both, since both are part of what is fundamentally occurring here.

On a purely physical level, what he is doing is:

Pushing them back while they hold on. This is reminiscent of a few of our escapes in Hapkido: Pushing them back weakens the grip and forces them to adjust their balance.

Pulling them down and to the side. This is pulling their center off, down, and to the right and aligning their (now forward) momentum along the angle to the side. This is particularly effective when combined with the previous technique, so that their balance is already disrupted.

By turning his arms over, it is forcing their arm all the way over to the opposite side from where it started.

As mentioned by others, there is a psychological force in play compelling everyone to go along with the technique (to be compliant, as it were). Part of this is the nature of the training and the situation (you want to look good for the cameras), part of this is actually practice (see below).

That's all part of the first component there. The second thing deals with how the uke are responding.

From here multiple things can result. If the uke does nothing here but just holds on tightly, then with a little extra movement in the technique the uke will probably end face-planting and/or ending up in a lock (depending on what the sensei is aiming for). Minimally, the sensei will end up in a position where his hands are free and he can strike his (hopefully off-balance) attacker.

Instead, they are going along with the technique, which trains them in how to escape it. This is part of the "compliance" piece: in hapkido we don't just train how to put the techniques on, but how to get out of them or keep the lock/throw from going into place on us. That's the other side of what you are seeing here: they are training to essentially break the lock and escape by kipping out of it. As we like to say: "Don't be the dummy for the other side."

This is further exaggerated by the desire to make it look good for the cameras, so you get what looks like a fluid and pretty much choreographed movement, where the instructor is slowly demonstrating a technique and students–who are training in the movement that will help them not get caught by it and have seen it many times before–see what is coming and go along with it.

I retract my answer - I have now viewed the technique, and I don't see ryoto mochi sukui nage (as done in tomiki's koryu dai san no kata section A) anywhere in that technique. I had to look up the Japanese, but ryoto mochi sukui nage is "two handed scooping technique"; my answer is limited to that throw.

At around the 9:21 mark of the video referenced in @Trevoke's question there is a two hand grab that seems to finish with some sort of teshi nage (heaven and earth throw), but when I say "some sort of", I really mean " I don't recognize it, but if you held a gun to my head and forced me to give it a name, I'd call it..."

---------------------- Old answer--------------------
(I haven't deleted it because I want to pursue the question raised by @DaveLiepman; I've edited the first line because I no longer believe the technique is related to OP).
There is a technique called ryoto mochi sukui nage. (approximately the 4 minute mark in that video).

As others have said, one of the keys is to interrupt the attack (sen sen no sen). If you allow them to take full initiatve and press the entire body weight on you, it is less likely to work.
A second key is to drop the elbows, turn the wrists inward and upwards so that the palms face one another and raise the wrists so that you apply the technique at the right angle. Dropping the elbows and rotating and raising the wrists has the effect of locking their elbows. uke cannot adjust their force if their elbows are locked.
The third and probably most important is to press upwards with the hips - as our sempai says, "The Pelvic thrusts are what drive them......" The hip thrust comes from your core and uses some of the strongest muscles in your body. The thrust is also directed upwards through the locked elbows, which (and I can't explain precisely how) compromises uke's structure. It is easier for Uke to rise than to endure hyperextended elbows.

As Uke rises, once again the locked elbows prevent uke from recovering balance. Like most of aikido, once you've stolen uke's balance, you have a variety of options. The turn and throw is the easiest and safest for both. There are variations of this that end in a tenshi nage as well.

If nage turns wrists inward and upwards, then if uke unlocks the elbows, uke is unable to exert any force on nage. To unlock the elbows, uke has to disengage from nage, which nullifies the attack. (The goal is to nullify the attack, not to complete the throw.) I'm struggling to write that more clearly - I'm not sure that explanation passes muster. I can no longer get into seiza, but perhaps I can persuade a classmate to work with it this weekend, and perhaps I'll find a clearer way to explain it.
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Mark C. WallaceMay 23 '13 at 18:06

Upvoting @DaveLiepmann's question, because I think it is insightful. This is an interesting discussion because we're polarized, and I appreciate the effort to narrow the discussion to a very pragmatic, answerable question.
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Mark C. WallaceMay 23 '13 at 18:09

I don't dispute that turning the wrists inward and upwards is an effective way to counter a double wrist grab; that's a universal technique. But why all this stuff after that move? Why pretend that uke is going to stand up and do a forward roll? In what universe would uke not just lose their grip instead of taking ukemi? That's the part that makes me dubious about the usefulness of the practice: there's no control during the part of the technique where impressive things happen; it's entirely due to assuming uke will cooperate or act drunk.
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Dave LiepmannMay 23 '13 at 18:17

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I'm not sure whether you replied before I finished my edit. These are koryu, which means they aren't done to be pragmatic. (The pragmatic stuff is in the junana hon kata).(overstating for simplicity). Most koryu techniques make... contrived assumptions about ukes attack and intent, and if the uke isn't trained to actually follow that script, you get the phenomenon you described "jump for sensei". Koryu is like a sonnet. Sonnets are written within narrow, artificial guidelines, not for pure pragmatism. Koryu are designed to teach concepts, not pragmatism. (IMHO)
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Mark C. WallaceMay 23 '13 at 18:24

There is no 'wave' sent into the opponent. The senior man is waving his hands and his students, knowing the drill, take a breakfall.

"Jump for sensei"

Many factors contribute to this being a "work": Okamoto is working with incredibly dedicated, long-term students of his. They are demonstrating for a crowd. They are being taped. Both uke and tori have invested years, if not decades, in making these techniques look effective on a compliant partner. In the past, it is almost certain that if Sensei attempted a technique and uke did not flip over and tap, Sensei would apply A) some other technique with a greater degree of force, and B) group social shaming of that individual for not receiving the technique compliantly.

Not everyone treats their martial art like a religion, but another, more inevitable problem for martial arts epistemology is that those who already have beliefs in the area tend to have a lot invested in those beliefs. The people whose testimony we are most likely to believe have inevitably put years of effort into perfecting their techniques.

More than anything else, Okamoto has set up a contrived parlor trick. He twitches his hands and the students jump for him. This is meant to showcase some broader principle or efficacy of technique, but largely it demonstrates the power of social suggestion and long-term setting of expectations.

For an example of how this fares against actual martial techniques, please see this challenge match (YouTube) of a "kiai master" against someone trained in punching and kicking.

Drilling for intangibles?

There's nothing wrong with odd drills that develop "intangibles". Balance, intuition, sensitivity, and timing are all hard to isolate during practice. Sometimes we'll do an odd drill that involves taking easy falls, but that should only be occasional. The more these drills (like the demonstrated Daito technique) supplant sparring and straightforward technique drilling, the more one should become dubious.

One should also be wary of techniques that seem to have over-the-top results. Sometimes, the technique is valid and just unusual. Other times, the technique simply does not have a solid basis. Instead, its production of over-the-top ukemi relies on cooperation from a compliant demonstration partner. Ask yourself: can this person execute this technique in chaotic conditions (like sparring) against opponents that aren't senior students?

Demonstration, not technique?

The most charitable interpretation one can give of this move is that this is a demonstration meant to illustrate a concept. It is not intended to be a technique that can be applied as shown and it is not intended to be interpreted as an effective combat technique. It's more Harlem Globetrotters than the Bulls.

The problem with such an approach is that nearly everyone who watches this demonstration fixates on the finger instead of the moon the finger is pointing at. People see this amazing "technique" and practice getting the demonstration to work instead of practicing fighting. It would be like someone training for the NBA spending more time on spinning a basketball on their finger than dribbling, passing, and shooting.

Demonstrations of concepts aren't so bad when the people doing it understand that they're engaging in historical re-enactment or cultural preservation. The problem is that most of those people think they're training to get better at fighting or defending themselves, and they're not. They're rehearsing a demo. It's a parlor trick used in the service of pedagogy that almost always fails to impart its lesson.

I would love to agree with you. Unfortunately, I have felt this done to me, had no idea was going to happen, had never seen this before, and it worked.
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AnonMay 14 '12 at 19:16

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The description above is accurate, but only partial in its explanation and insight. Any technique in application is only part of the picture; it may or may not work, so the artist is prepared to perform another movement or adjustment to create an effect. This answer is of relatively low quality since the assumptions made are not grounded in anything other than speculation and bias.
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stslavikMay 14 '12 at 20:21

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I'm with Dave on this, I think this is a good answer!
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Keith NicholasMay 15 '12 at 0:48

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I was not expecting a particular technique to be performed, and I certainly did not want to please anyone. How is this consistent with your analysis that the uke is eager to please?
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AnonMay 15 '12 at 19:37

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Looking at the video in full (and not just the last technique), I am afraid that you are spot on: this is falling for sensei. My apologies for down voting. However, the last technique can be made to work by locking the arms and moving uke. The rest...
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SardathrionJun 2 at 13:04

It is hard to explain what is shown without writing a book on aiki work and anatomy. Like most other things on this site, it must be felt. My words, therefore, will sadly lack a certain precision. If this explanation is interesting to you, but feels incomplete, track down an aikijujutsu school.

The first thing to realize is that what we are looking at is aiki work. In order to aspire to understand this, one must be aware of the interactions between the elements of the body's anatomy, such as tendons and fascia, in addition to the more obvious bones and muscles. Here is a diagram showing layers in the human body. This is your new world.

The second thing to understand is that all the movements come from the core of the body and are applied in every direction. Most martial artists eventually realize the futility of getting very strong arms if they can just apply the power from the much stronger core muscles, but almost all training remains unidirectional (just apply energy in that direction, then step in that direction). This is different, and is usually referred to as expressing in the six directions.

The first technique we see is called aiki age (rising energy). The second technique, the "throw" is mostly inconsequential to this conversation, although it also requires a large amount of training.

Tori has to press forward and apply a subtle and definite kuzushi, which is just a Japanese term to mean unbalance. A direct consequence is that uke is not grounded and therefore is unable to resist effectively when forces are applied (e.g. try applying a punch to someone when they stand on the tip of their toes).

A requirement of the proper application of aiki is to seal the opponent's breath by preventing the muscles around the lungs from properly expanding and contracting. The sudden application of this can be heard in the surprised and pained screams. This is different from being hit in the solar plexus because it's not a shock trauma, more like a very sudden squeeze. It's the difference between punching a towel and twisting it. I do not know exactly what causes this to happen inside uke's body.

The art of aiki comes into understanding which directions and vectors to use for maximum efficiency. In this video, for instance, after kuzushi is applied, vectors are applied into uke's body, turning and locking the scapulas while arching them backwards, preventing uke from recovering in any way. Uke is now locked in a position of unbalance.

The second technique is the art of directing uke from their locked position. It is a matter of sending the right combination of vectors in the right chronology through both hands. A similar technique is shown here between 17:02 and 17:14, where the technique is demonstrated, then shown without an opponent. The difference is that here it is applied through the shoulders, not the hands.

The vector must be applied with the entire body acting in concert. If it is not done so, it will not have the desired effect. Pushing or pulling, for instance, will cause a localized change in uke's body, and we are looking for a global change. This is why the entire body must act in concert.

Here is a video showing a simple demonstration / explanation (in Japanese) of aiki age which relies on muscle/bone alignment (that is, a lower-level version of what is shown in the original video).

Here is a video (in English) of an aikidoka showing aiki age + aiki sage (rising + descending energy). The explanation is relatively good, though not necessarily as detailed as one who might wish.